Spatial Interpolation Methods

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2018

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1. Introduction

Spatial interpolation is the procedure to predict the value of attributes at unobserved points within a study region using existing observations (Waters, 1989). Lam (1983) definition of spatial interpolation is “given a set of spatial data either in the form of discrete points or for subareas, find the function that will best represent the whole surface and that will predict values at points or for other subareas”.

Predicting the values of a variable at points outside the region covered by existing observations is called extrapolation (Burrough and McDonnell, 1998). All spatial interpolation methods can be used to generate an extrapolation (Li and Heap 2008).

Spatial Interpolation is the process of using points with known values to estimate values at other points. Through Spatial Interpolation, We can estimate the precipitation value at a location with no recorded data by using known precipitation readings at nearby weather stations.

Rationale behind spatial interpolation is the observation that points close together in space are more likely to have similar values than points far apart (Tobler’s Law of Geography).

Spatial Interpolation covers a variety of method including trend surface models, thiessen polygons, kernel density estimation, inverse distance weighted, splines, and Kriging.

Spatial Interpolation requires two basic inputs:

· Sample Points

· Spatial Interpolation Method

Sample Points

Sample Points are points with known values. Sample points provide the data necessary for the development of interpolator for spatial interpolation.

The number and distribution of sample points can greatly influence the accuracy of spatial interpolation. 2. Spatial Interpolation Methods

Point interpolation methods may be categorized in several ways.

· local or global · exact or approximate

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· gradual or abrupt · deterministic or stochastic

Global Interpolation uses every known point available to estimate an unknown value. Local Interpolation uses a sample of known points to estimate an unknown value. This method is designed to capture the local or short range variation.

Exact interpolation predicts a value at the point location that is the same as is known value.

Approximate interpolation (inexact) predicts a value at the point location that differs from its known value.

Deterministic interpolation method provides no assessment of errors with predicted values. Stochastic interpolation methods offer assessment of prediction errors with estimated variance.

Gradual or abrupt interpolators are distinguished by the continuity of the surface they produce. Gradual interpolation techniques will produce a smooth surface with gradual changes occurring between observed data points.

A wide variety of spatial interpolation methods exist in the literature. The spatial interpolation methods covered in this review are only those commonly used in the studies.

2.1.Theissen polygons

Theissen polygons are an exact method of interpolation that assumes the unknown values of the points on a surface to be equal to the value of the nearest known point. This can be considered as a local interpolator because the characteristics of the global data set have no influence on the interpolation process.

The steps to follow are:

1. Construct the Thiessen Polygons

2. Determine the area proportions of each polygon and assign a weight

3. Calculate the average value by using the weights (Thiessen, 1911)

The method of Theissen polygons is a robust technique and will always produce the same surface from the same set of data points. This is a disadvantage that the technique is unintelligent and unable to respond to external knowledge about factors which may influence the values recorded at the observed data points. One of the most common uses for this technique is for the generation of area territories from a set of points.

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2.2.The Triangulated Irregular Network (TIN)

The Triangulated Irregular Network (TIN) developed in the early 1970’s is a simple way to construct a surface from a set of known points. It is a particularly useful technique for irregularly spaced points. The TIN approach is an exact interpolation method. In the TIN model, the known data points are connected by lines to form a series of triangles.

A TIN is a terrain model type that creates a set of continuous, non-overlapping, connected triangles (faces), based on a so-called Delaunay triangulation of irregularly spaced observations. The corners of the triangles are identical to the observations and within each triangle the surface is usually represented by a plane. A TIN model will allow for different density in the data model in different areas. The size of the triangles can therefore be adjusted to reflect the degree of relief in the surface to be modelled, provided more data has been gathered in areas of variable elevation characteristics. The use of triangles ensures that each piece of the surface fits the neighbouring pieces, and it is thus possible to create the desired continuous seamless transition between the triangles, within the model.

2.3. Trend Surface Models

An approximate method, trend surface analysis approximates points with known values with a polynomial equation.

A surface interpolation method that fits a polynomial surface by least-squares regression through the sample data points. This method results in a surface that minimizes the variance of the surface in relation to the input values. The resulting surface rarely goes through the sample data points. This is the simplest method for describing large variations, but the trend surface is susceptible to outliers in the data. Trend surface analysis is used to find general tendencies of the sample data, rather than to model a surface precisely (ESRI).

The linear equation or the interpolator can then be used to estimate values at other points.

푍(푥, 푦) = 푏0+푏1푥 + 푏2푦

Where the attribute value z is a function of x and y coordinates. The b coefficients are estimated from the known points.

The quadratic interpolator can be written as given below.

2 2 푍(푥, 푦) = 푏0+푏1푥 + 푏2푦 + 푏3푥 + 푏4푦 + 푏5푥푦 Reference # Page | 4 of 4

The two general classes of techniques for estimating a regular grid of points on a surface from scattered observations are methods called "global fit" and "local fit." As the name suggests, global- fit procedures calculate a single function describing a surface that covers the entire map area. The function is evaluated to obtain values at the grid nodes. In contrast, local-fit procedures estimate the surface at successive nodes in the grid using only a selection of the nearest data points.

Trend surface analysis is the most widely used global surface-fitting procedure. The mapped data are approximated by a polynomial expansion of the geographic coordinates of the control points, and the coefficients of the polynomial function are found by the method of least squares, insuring that the sum of the squared deviations from the trend surface is a minimum. Each original observation is considered to be the sum of a deterministic polynomial function of the geographic coordinates plus a random error.

The polynomial can be expanded to any desired degree, although there are computational limits because of rounding error. The unknown coefficients are found by solving a set of simultaneous linear equations which include the sums of powers and cross products of the X, Y, and Z values. Once the coefficients have been estimated, the polynomial function can be evaluated at any point within the map area. It is a simple matter to create a grid matrix of values by substituting the coordinates of the grid nodes into the polynomial and calculating an estimate of the surface for each node. Because of the least-squares fitting procedure, no other polynomial equation of the same degree can provide a better approximation of the data.

The trend surface method fits a surface to the known observation points by a method known as least squares. The shape of the surface is determined by the mathematical equation, or polynomial, which is used to describe it. A polynomial surface has some very important characteristics; most importantly it changes smoothly and predictably from one map edge to the other. The polynomial equation may be linear, quadratic or even cubic. A linear equation would be used to describe a tilted plane. A quadratic equation would define a simple hill or hollow and a cubic equation would define a more complex surface. The trend surface method fits a surface to the known observation points by a method known as least squares. The shape of the surface is determined by the mathematical equation, or polynomial, which is used to describe it. A polynomial surface has some very important characteristics; most importantly it changes smoothly and predictably from one map edge to the other. The polynomial equation may be linear, quadratic or even cubic. A linear equation would be used to describe a tilted plane. A quadratic equation would define a simple hill or hollow and a cubic equation would define a more complex surface.

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2.4. Nearest neighbor interpolation

Nearest-neighbor interpolation is a simple method of multivariate interpolation in one or more dimensions.

Interpolation is the problem of approximating the value of a function for a non-given point in some space when given the value of that function in points around (neighboring) that point. The nearest neighbor algorithm selects the value of the nearest point and does not consider the values of neighboring points at all, yielding a piecewise-constant interpolant. The algorithm is very simple to implement and is commonly used (usually along with mipmapping) in real-time 3D rendering to select color values for a textured surface.

For a given set of points in space, a Voronoi diagram is a decomposition of space into cells, one for each given point, so that anywhere in space, the closest given point is inside the cell. This is equivalent to nearest neighbor interpolation, by assigning the function value at the given point to all the points inside the cell. The figures on the right side show by color the shape of the cells.

Nearest neighbor is the most basic technique and as it requires the least processing time which is major advantage among all the interpolation algorithms because it takes only one pixel into consideration i.e. the nearest one to the interpolated point. This has the results into simply making each pixel larger. Although nearest neighbor method is very efficient because of its time efficiency, the quality of image is comparatively very poor.

2.5. Natural neighbor interpolation

Natural neighbor interpolation is a method of spatial interpolation, developed by Robin Sibson (1980, 1981). The method is based on Voronoi tessellation of a discrete set of spatial points. This has advantages over simpler methods of interpolation, such as nearest-neighbor interpolation, in that it provides a smoother approximation to the underlying "true" function.

The points used to estimate the value of an attribute at location x are the natural neighbors of x, and the weight of each neighbor is equal to the natural neighbor coordinate of x with respect to this neighbor. If we consider that each data point in S has an attribute ai (a scalar value), the natural neighbor interpolation is given below.

n f(x)   wi (x)ai i1

Where f(x) is the interpolated function value at the location x. The resulting method is exact and f(x) is smooth and continuous everywhere except at the data points.

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2.6. The Moving Average Method

The Moving Average method assigns values to grid nodes by averaging the data within the grid node's search ellipse. To use Moving Average, a search ellipse must be defined and the minimum number of data to use, specified. For each grid node, the neighboring data are identified by centering the search ellipse on the node. The output grid node value is set equal to the arithmetic average of the identified neighboring data. If there are fewer, than the specified minimum number of data within the neighborhood, the grid node is blanked.

This process involves calculating a new value for each known location based upon a range of values associated with neighboring points. The new value will usually be either an average or weighted average of all the points within a predefined neighborhood. In some textbooks you will find that they use the term ‘moving window’.

Applying a moving average involves making a number of decisions about the shape, size and character of the neighborhood about a point. The most common shape for a neighborhood is a circle, since points in all directions have an equal chance of falling within the radius of a given point. However, in the raster world a square or rectangular window of cells is often used. The size of the neighborhood is determined by the user and is usually based upon assumptions about the distance over which local variability in the data is important.

2.7. Inverse Distance Weighting (IDW)

It is an exact method that enforces the condition that the estimated value of a point is influenced more by nearby known points than by those farther away.

The IDW is simple and intuitive deterministic interpolation method based on principle that sample values closer to the prediction location have more influence on prediction value than sample values farther apart. Using higher power assigns more weight to closer points resulting in less smoother surface. On the other hand, lower power assigns low weight to closer points resulting in smoother surface. Major disadvantage of IDW is “bull's eye” effect (higher values near observed location) and edgy surface. The process is highly flexible and allows estimating dataset with trend or anisotropy, in search neighborhood shaping (Burrough and McDonnel, 1988; Liu, 1999). Reference # Page | 7 of 7

IDW combines the notion of proximity with that of gradual change of the trend surface. IDW relies on the assumption that the value at an unsampled location is a distance-weighted average of the values from surrounding data points, within a specified window. The points closest to the prediction location are assumed to have greater influence on the predicted value than those further away, such that the weight attached to each point is an inverse function of its distance from the target location.

The general IDW prediction formula is

N ˆ Z(u0 )   wi Z(ui ) i1 where: Z(u0) is the value being predicted for the target location u0; N is the number of measured data points in the search window; wi are the weights assigned to each measured point; and Z(ui) is the observed value at location ui. ui=(xi,yi.)

1/ di wi  N 1/ di i1

d 2  (x  x )2  (y  y )2 i i i The Inverse Distance Method is very simple and easy to use which is one of its biggest advantages. It is applicable for a wide range of data as the method often delivers reasonable results and does not exceed the range of meaningful values (Caruso et al., 1998).

2.8. Spline

Spline is deterministic interpolation method which fits mathematical function through input data to create smooth surface. Spline can generate sufficiently accurate surfaces from only a few sampled points and they retain small features. Spline works best for gently varying surfaces like temperature.

Splines are piece-wise polynomial functions that are fitted together to provide a complete, yet complex, representation of the surface between the measurement points. Functions are fitted exactly to a small number of points while, at the same time, ensuring that the joins between different parts of the curve are continuous and have no disjunctions. Splines are often useful for calculating smooth surfaces from a large number of input data points and often produce good results for gently varying surfaces such as elevation. Thin plate splines are a special form of spline, which have particular value in interpolation. These, in effect, replace the exact spline surface with a locally smoothed average which passes as close as possible through the data points. They can be Reference # Page | 8 of 8 used to remove artifacts (e.g. excessively high or low predicted values) resulting from natural variation and measurement error in the input data.

Spline creates a surface that passes through the control points and has the least possible changes in slope at all the points.

N 2 Q(x, y)   Ai di logdi  a  bx  cy i1

d 2  (x  x )2  (y  y )2 i i i

Where x and y are the coordinates of the point to be interpolated, and xi and yi are the coordinates of control point.

Unlike the IDW method, the predicted values from Spline are not limited within the range of maximum and minimum values of the known points.

Instead of averaging values, the Spline interpolation method fits a flexible surface, as if it were stretching a rubber sheet across all the known point values. Cliff face or a fault line, are not represented well by a smooth-curving surface. In such cases, you might prefer to use IDW interpolation, where barriers can be used to deal with these types of abrupt changes in local values. There are two types of Spline.

A Regularized Spline yields a smooth surface and smooth first derivatives. With the Regularized option, higher values used for the [weight] parameter produce smoother surfaces. The values entered for this parameter must be equal to or less than zero. Typical values that may be used are 0, 0.001, 0.01, 0.1, and 0.5.

A Tension Spline is flatter than a Regularized Spline of same sample points, forcing the estimate to stay closer to the sample data. It produces a surface more rigid according to the character of the modeled phenomenon.

With the Tension option, higher values entered for the [weight] parameter result in somewhat coarser surfaces, but surfaces that closely conform to the control points. The values entered must be equal to or greater than zero. The typical values are 0, 1, 5, and 10.

2.9.Kriging

Kriging, also known as the “Theory of regionalized variables”, was developed by G. Matheron and D. G. Krigge as an optimal method of interpolation for use in the mining industry. Burrough (1986) Reference # Page | 9 of 9 notes that the origins of the method lie in the recognition that the spatial variation of many geographical properties is too irregular to be modeled by a smooth mathematical function.

The basis of kriging lies in estimating the average rate at which the difference between values at points change with distance between points. This is the most complex of all the exact interpolation methods and until recently was absent from virtually all commercially available GIS software.

Kriging was developed in the 1960s by the French mathematician Georges Matheron. The motivating application was to estimate gold deposited in a rock from a few random core samples. Kriging has since found its way into the earth sciences and other disciplines. It is an improvement over inverse distance weighting because prediction estimates tend to be less bias and because predictions are accompanied by prediction standard errors (quantification of the uncertainty in the predicted value).

The basic tool of geostatistics and kriging is the semivariogram. The semivariogram captures the spatial dependence between samples by plotting semivariance against separation distance

Geostatistics assumes spatial data analysis. The most common spatial tool is variogram defined by formula given below.

N (h) 1 2  (h)   z(ui )  z(ui  h) 2N(h) i1

N(h)is number of data pairs at distance “h”, z(ui)is value at location ui=(xi,yi) and z(ui+h) value at location ui+h

Calculation of experimental variogram is necessary input for different geostatistical interpolation or simulation techniques, like kriging given below.

n z(uo )  i  z(ui ) i1 uo is points estimated by kriging, i is- weight coefficient for kriging calculated from matrix equation and z(ui ) is hard data of variable.

The kriging includes several interpolation techniques like Simple, Ordinary, Universal and other kriging interpolations.

For more detailed information the classical textbooks by Creessie (1993) and Journel & Huıjbregts (1978) are the important references.

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2.9.1 Simple Kriging

Model assumptions of Z(u) are given below.

푍(푢) = 휇 + 휀(푢)

퐸[휀(푢)] = 0

퐸(푍(푢)) = 휇 constant and known

Simple kriging predictor for the point uo is given below.

푛

푍̂(푢0) = ∑ 휆푖 푍(푢푖) 푖

휆 = Г−1훾

′ 훾 = (훾(푢1 − 푢표), . . , 훾(푢푛 − 푢표))

′ 휆 = (휆1, 휆2, 휆3, … . , 휆푛)

훾(푢1 − 푢1) ….. 훾(푢1 − 푢푛)

Г = …. ….. …..

훾(푢푛 − 푢1) ….. 훾(푢푛 − 푢푛)

2.9.2. Ordinary Kriging

Model assumptions of Z(u) are given below.

푍(푢) = 휇 + 휀(푢)

퐸[휀(푢)] = 0

퐸(푍(푢)) = 휇 constant and unknown

Ordinary kriging predictor for the point uo is given below.

푛

푍̂(푢0) = ∑ 휆푖 푍(푢푖) 푖

휆′ = (훾 + 1(1 − 1′Г−1훾)/1′Г−11)′Г−1

1=(1,…,1) n dimensional unit vector.

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2.9.2 Universal Kriging

Model assumptions of Z(u) are given below.

푍(푢) = 휇(푢) + 휀(푢)

퐸[휀(푢)] = 0

푏 휇(푢) = ∑푗=0 훽푗푓푗(푢) and 푓1(푢), 푓2(푢), … , 푓푏(푢)

Universal kriging predictor for the point uo is given below.

푛

푍̂(푢0) = ∑ 휆푖 푍(푢푖) 푖

−1 휆′ = (훾 + 퐹(퐹′Г−1퐹) (푓 − 퐹′Г−1훾))′Г−1

f1(u1) … fb(u1)

퐹 =

f1(un) … fb(un)

푓1(푢) = ( 푓1(u0), … , 푓푏(u0)

Reference # Page | 12 of 12 3. Application of spatial interpolation methods on an example

Sample data given in Table 1 was produced randomly with space filling experimental design to implement the interpolation methods discussed above. Column x and column y show the coordinates and the column z shows the random variable (or attribute) of the point.

Table 1. Sample data of 100 spatial points

Point x y z Point x y z Point x y z Point 1 1 13 76.00 Point 35 39 1 196.00 Point 69 72 29 147.00 Point 2 3 68 167.00 Point 36 39 15 227.00 Point 70 73 3 182.00 Point 3 4 30 239.00 Point 37 41 43 136.00 Point 71 73 34 179.00 Point 4 4 86 116.00 Point 38 42 76 209.00 Point 72 73 48 182.00 Point 5 5 1 124.00 Point 39 42 78 140.00 Point 73 74 54 130.00 Point 6 5 78 87.00 Point 40 45 57 89.00 Point 74 74 85 159.00 Point 7 7 10 191.00 Point 41 46 35 171.00 Point 75 75 31 62.00 Point 8 8 47 78.00 Point 42 46 65 60.00 Point 76 76 26 137.00 Point 9 10 58 67.00 Point 43 46 67 61.00 Point 77 79 47 71.00 Point 10 11 3 70.00 Point 44 46 81 168.00 Point 78 81 97 126.00 Point 11 13 60 185.00 Point 45 47 30 135.00 Point 79 82 72 138.00 Point 12 14 13 102.00 Point 46 50 89 105.00 Point 80 83 96 202.00 Point 13 14 63 129.00 Point 47 53 5 144.00 Point 81 85 12 164.00 Point 14 15 1 124.00 Point 48 54 52 103.00 Point 82 85 69 99.00 Point 15 15 87 227.00 Point 49 55 26 120.00 Point 83 85 89 90.00 Point 16 18 16 172.00 Point 50 56 1 191.00 Point 84 87 42 214.00 Point 17 19 37 145.00 Point 51 56 41 98.00 Point 85 88 83 160.00 Point 18 20 17 231.00 Point 52 56 62 91.00 Point 86 91 18 184.00 Point 19 20 35 208.00 Point 53 57 35 178.00 Point 87 91 92 225.00 Point 20 20 80 232.00 Point 54 60 59 135.00 Point 88 92 42 199.00 Point 21 21 52 75.00 Point 55 60 60 100.00 Point 89 92 54 169.00 Point 22 21 63 157.00 Point 56 61 2 166.00 Point 90 92 84 137.00 Point 23 22 57 127.00 Point 57 61 19 167.00 Point 91 92 91 126.00 Point 24 23 23 245.00 Point 58 61 30 80.00 Point 92 93 41 158.00 Point 25 24 51 124.00 Point 59 62 6 187.00 Point 93 93 88 181.00 Point 26 25 60 136.00 Point 60 62 59 79.00 Point 94 95 31 176.00 Point 27 29 18 86.00 Point 61 62 81 76.00 Point 95 96 31 164.00 Point 28 30 85 175.00 Point 62 65 94 149.00 Point 96 97 41 193.00 Point 29 31 4 119.00 Point 63 66 11 144.00 Point 97 97 95 130.00 Point 30 32 38 121.00 Point 64 66 27 58.00 Point 98 99 36 99.00 Point 31 32 53 238.00 Point 65 66 36 87.00 Point 99 99 69 99.00 Point 32 32 76 54.00 Point 66 67 61 73.00 Point 100 100 83 103.00 Point 33 33 29 115.00 Point 67 69 11 134.00 Point 34 35 17 203.00 Point 68 71 75 71.00

100 sample grids are randomly selected from 10000 grids. Reference # Page | 13 of 13

Map 1. Display of 100 observation point from a 100x100 grid map

Table 2. Sample statistics

Statistics Value

Mean 140,58

Median 136,5

Variance 2493,317

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Map 2. Display of trend surface interpolation map (100x100 grid)

Table 3. Statistics of trend surface interpolation

Statistics Value

Mean 139,76

Median 139,7

Std.Dev. 6,74

Prediction value 1397580 (10000 grids)

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Map 3. Display of nearest point interpolation map (100x100 grid)

Table 4. Statistics of nearest point interpolation

Statistics Value

Mean 141,84

Median 137

Std.Dev. 51,26

Prediction value 1484000 (10000 grids)

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Map 4. Display of linear moving average interpolation map (100x100 grid)

Table 5. Statistics of linear moving average interpolation

Statistics Value

Mean 142,31

Median 141,7

Std.Dev. 21

Prediction value 1423100 (10000 grids)

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Map 5. Display of moving surface interpolation map (100x100 grids)

Table 6. Statistics of moving surface interpolation

Statistics Value

Mean 144,19

Median 141,6

Std.Dev. 34,99

Prediction value 1441900 (10000 grids)

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Map 6. Display of ordinary kriging interpolation map (100x100 grid)

Table 7. Statistics of ordinary kriging

Statistics Value

Mean 141,92

Median 142

Std.Dev. 23,98

Prediction value 1419200 (10000 grids)

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Map 7. Display of universal kriging (linear trend) interpolation map (100x100 grid)

Table 8. Statistics of universal kriging (linear trend)

Statistics Value

Mean 144,48

Median 143,94

Std.Dev. 26,58

Prediction value 1448000 (10000 grids)

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Map 8. Display of universal kriging (quadratic trend) interpolation map (100x100 grid)

Table 9. Statistics of universal kriging (quadratic trend)

Statistics Value

Mean 144,74

Median 143,98

Std.Dev. 34,82

Prediction value 1447400 (10000 grids)

Reference # Page | 21 of 21 4. Conclusions

Interpolation is a method that predicts the values at locations where no sample values are available. Spatial interpolation assumes the attribute data are continuous over space. This allows the prediction of the attribute at any location within the data boundary. Another assumption is the attribute is spatially dependent, indicating the values closer together are more likely to be similar than the values farther apart. These assumptions allow for the spatial interpolation methods to be formulated.

In many areas spatial interpolation methods have been used successfully such as geology, environment, mining, climatology, biology, forestry, agriculture, engineering etc.

Spatial interpolation provides comprehensive information about the spatial attributes (or variables) of any interested subject for points or grids.

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